Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,11)(7,15)(8,9)(12,14)(16,17)(18,19);(5,6,8,11,10,14,9,12)(7,13,15,18,17,20,16,19);(2,4)(6,12)(7,16)(8,9)(11,14)(13,20)(15,17);(5,7)(6,13)(8,15)(9,16)(10,17)(11,18)(12,19)(14,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238734 |
| maximal normal subgroup of | #72238734 |
| maximal quotient of | #72238738, #72238739 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371856);